1. Traffic Grooming in WDM Ring Networks Presented by: Eshcar Hilel

2. 236357 - Distributed Algorithms, Spring 2005 2 Introduction Optical Networks - A new generation of networks using optical fiber transmission – Excellent medium, high BW, low error … SONET ring - synchronous optical network, currently the most widely deployed optical network infrastructure WDM Technology – wavelength-division multiplexing

3. 236357 - Distributed Algorithms, Spring 2005 3 Introduction – SONET Ring SADM - SONET add/drop multiplexers can aggregate lower-rate signals into a single high- rate stream SONET ring use one fiber pair (or two for protection) to connect SADMs in the source and destination nodes

4. 236357 - Distributed Algorithms, Spring 2005 4 Introduction – WDM Increases the transmission capacity of optical fibers Allows simultaneously transmission of multiple wavelengths (channels) within a single fiber One wavelength may carry Internet traffic; another may carry voice or video

5. 236357 - Distributed Algorithms, Spring 2005 5 Introduction – SONET over WDM Multiple SONET rings can be supported on a single fiber pair by using multiple wavelengths The networks are limited by the processing capability of electronic switches, routers and multiplexers (not by transmission bandwidth) New aim: overcoming the electronic bottleneck by providing optical bypass

6. 236357 - Distributed Algorithms, Spring 2005 6 Introduction – Optical bypass WADM - WDM Add/Drop Multiplexer allows to drop (or add) only the wavelength that carries the traffic destined to (or originated from) the node The dropped wavelength is electronically processed at the node All the other wavelengths optically bypass the node

7. 236357 - Distributed Algorithms, Spring 2005 7 Introduction – WADM More optical switches may be added to support more add-drop wavelengths

8. 236357 - Distributed Algorithms, Spring 2005 8 Introduction – Traffic Grooming Every wavelength needs a SADM only at nodes where it is ended Traffic typically require only a small fraction of the wavelength Traffic grooming can be used in such a way that all of the traffic to and from the node is carried on minimum number of wavelength

9. 236357 - Distributed Algorithms, Spring 2005 9 Topics of Discussion Traffic Grooming - Understanding the Problem Single Exit Node Network – NP-complete problem – Special case: uniform traffic – Special case: minimum number of wavelengths All-To-All Uniform Traffic Network

10. Traffic Grooming Understanding the problem

11. 236357 - Distributed Algorithms, Spring 2005 11 What ’ s the Problem? Unidirectional (clockwise) WDM ring N nodes: 1,2, …,N c – grooming factor r ij - number of low rate circuits from node i to node j Objective: minimize total number of SADMs

12. 236357 - Distributed Algorithms, Spring 2005 12 Illustration Unidirectional ring network: N = 4 6 pairs of nodes r ij = 8: 8 OC-3 circuits between each pair c = 16: each wavelength supports an OC-48 ring Total load: 6x8 OC-3 = 3 OC-48, requires 3 wavelengths

13. 236357 - Distributed Algorithms, Spring 2005 13 Illustration Traffic assignment: 1: 1↔2, 3↔4 2: 1↔3, 2↔4 3: 1↔4, 2↔3 Total: 12 SADMs

14. 236357 - Distributed Algorithms, Spring 2005 14 Illustration Traffic assignment: 1: 1↔2, 1↔3 2: 2↔3, 2↔4 3: 1↔4, 3↔4 Total: 9 SADMs

15. 236357 - Distributed Algorithms, Spring 2005 15 Goal – Traffic Grooming Tradeoff between efficient use of fibers and the cost of electronic equipment When no limitation on wavelengths – dedicated wavelength per connection, no multiplexing Else design traffic grooming algorithms to – Minimize number of electronics (SADMs) – Minimize number of wavelengths (efficient use of wavelengths)

16. Single Exit Node Network E. Modiano, A. Chio, “ Traffic Grooming Algorithms for Reducing Electronic Multiplexing Costs in WDM Ring Networks ”

17. 236357 - Distributed Algorithms, Spring 2005 17 Telephone company’s central office Computational Complexity Unidirectional ring All the traffic on the ring is destined to a single exit node Denote the exit node 0 r ij > 0, for j = 0 and i = 1, …,N Note: maximum load L max =  i=1..N r i0 and minimum wavelengths W min =  L max / c 

18. 236357 - Distributed Algorithms, Spring 2005 18 Computational Complexity Assume w.l.o.g. r i0 <c for all i Else fill  r i0 /c  wavelengths with  r i0 /c  *c low rate circuits, and groom the remaining (<c) circuits Theorem: The traffic grooming problem is NP-complete

19. 236357 - Distributed Algorithms, Spring 2005 19 Computational Complexity Bin packing problem: What is the least number of bins (containers of fixed volume) needed to hold a set of objects (of different volumes)? The bin packing problem is an NP-complete problem.

20. 236357 - Distributed Algorithms, Spring 2005 20 Computational Complexity Claim: There exist an optimal solution such that no traffic from a node is split onto two rings Proof: – Consider assignment where the traffic of some nodes is split onto 2 or more rings – Each such node have at least 2 SADMs – Accommodate the traffic on a separate wavelength – Requires at most 2 SADMs

21. 236357 - Distributed Algorithms, Spring 2005 21 Computational Complexity Theorem Proof: – For any optimal solution with no split traffic: regular nodes - N SADMs; exit node - k SADMs, where k is the number of SONET rings – Problem reduced to minimizing total number of rings – Achieved by combining traffic from multiple nodes onto single ring (wavelength) – This is basically the Bin Packing problem! QED

22. 236357 - Distributed Algorithms, Spring 2005 22 Special Case: Uniform Traffic r i0 = r Optimal solution does not require split traffic May groom traffic from at most  c/r  nodes on one SONET ring Number of wavelengths: W =  N/  c/r   Hence, minimum SADMs M min = N + W Not the minimum number of wavelengths!

23. 236357 - Distributed Algorithms, Spring 2005 23 Special Case: Minimum Number of Wavelengths Traffic from nodes may have to be split onto multiple rings, S - total number of traffic splits Additional SADM per split Hence, #SADMs M = N + W min + S, where W min =  r*N /c  Objective: minimize the total number of splits

24. 236357 - Distributed Algorithms, Spring 2005 24 Special Case: Minimum Number of Wavelengths Maximum load for ring with no split L ns =  c/r  *r W ns Maximum number of rings with no split Remaining rings contain at most c circuits: W ns * L ns + (W min - W ns )*c >= L max W ns = min{W min,  (c* W min – L max ) / (c-L ns )  }

25. 236357 - Distributed Algorithms, Spring 2005 25 Iterative Algorithm Initialization: c 0 = c, N 0 = N, r 0 = r, W 0 = W 0 min Steps of loop i: – If W i ns = W i min then accommodate the remaining traffic without splitting - terminate – Fill W i rings with unsplit traffic from  c i /r i  nodes – Remaining capacity is c i+1 = c i -  c i /r i  *r i – N i+1 = N i -  c i /r i  *W i nodes needs to be assigned

26. 236357 - Distributed Algorithms, Spring 2005 26 Steps of loop i (cont): – N i+1 = N i -  c i /r i  *W i nodes needs to be assigned – Fill remaining capacity c i+1 by traffic from N i+1 nodes – Remaining traffic becomes r i+1 = r i – c i+1 – W i+1 = W i – N i+1 – Continue to loop i+1 N i+1 < W i Iterative Algorithm (cont)

27. All-To-All Uniform Traffic Network J.C. Bermond, D. Coudert, “ Traffic Grooming in Unidirectional WDM Ring Networks using Design Theory ”

28. 236357 - Distributed Algorithms, Spring 2005 28 All-To-All Uniform Traffic We show the problem can be formulated in terms of graph partition into sub-graphs: – at most c edges and per sub-graph – minimize total number of vertices

29. 236357 - Distributed Algorithms, Spring 2005 29 Traffic Grooming: Reformulating the Problem N nodes of unidirectional ring C N R = N(N-1)/2 circles c – grooming factor K N - Complete graph on N vertices B λ denote a sub-graph of K N V(B λ ) (resp E(B λ )) denote its vertex (resp edge) set

30. 236357 - Distributed Algorithms, Spring 2005 30 Traffic Grooming: Reformulating the Problem B λ correspond to a wavelength An edge of B λ correspond to a circle in the ring B λ is viewed as a set of circles packed in a wavelength |E(B λ )| <= c V(B λ ) correspond to the number of SADMs A(c,N) denotes total number of SADMs

31. 236357 - Distributed Algorithms, Spring 2005 31 Traffic Grooming: Reformulating the Problem Input:N and c Output:partition of K N into sub-graphs B λ, λ = 1, …,W, such that |E(B λ )| <= c Objective: minimize ∑ 1<=λ<=W |V(B λ )|

32. 236357 - Distributed Algorithms, Spring 2005 32 Lower Bound ρ(B λ ) = |E(B λ )|/|V(B λ )| is the sub-graph ratio ρ(m) maximum ratio of sub-graph with m edges ρ max (c) = max m<=c ρ(m)

33. 236357 - Distributed Algorithms, Spring 2005 33 Lower Bound Theorem: any grooming of R circles with grooming factor c needs at least R/ρ max (c) SADMs Proof: R = ∑ W λ=1 |E(B λ )| <= ρ max (c)* ∑ W λ=1 |V(B λ )| Thus we have the lower bound: A(c,N) >= N(N-1) / ρ max (c)*2

34. 236357 - Distributed Algorithms, Spring 2005 34 Lower Bound We compute ρ max (c) Theorem: If k(k-1)/2<=c<=(k+1)(k-1)/2, then ρ max (c)=(k-1)/2 If (k+1)(k-1)/2<=c<=(k+1)k/2, then ρ max (c)=c/k+1 Proof: on board

35. 236357 - Distributed Algorithms, Spring 2005 35 Lower Bound Note: these sub-graphs do not have necessarily exactly c edges and so the minimum is not necessarily attained for W = W min Example: N=13 and c=7

36. 236357 - Distributed Algorithms, Spring 2005 36 Discussion My opinion of the subject Your opinion of the subject (and presentation … ) That ’ s all folks!

37. 236357 - Distributed Algorithms, Spring 2005 37 References J.C. Bermond, D. Coudert, “ Traffic Grooming in Unidirectional WDM Ring Networks using Design Theory ”, IEEE International Conference on Communications, May, 2003 E. Modiano, A. Chio, “ Traffic Grooming Algorithms for Reducing Electronic Multiplexing Costs in WDM Ring Networks ”, IEEE J. Lightwave Tech., Jan. 2000 vol. 18(1)

38. 236357 - Distributed Algorithms, Spring 2005 38 References E. Modiano, P. Lin, “ Traffic Grooming in WDM Networks ”, IEEE Communication Magazine, July 2001.